In astrophysics there is a lot going on about strong, large scale magnetic fields: in stars (prominences), magnetic dynamos, compact accretors collimating jets, etc. There's even a special computational formalism called magnetohydrodynamics (MHD), which allows to deal with space plasma.

Yet I've never read about large scale electric fields. I know that most of the matter we model in astrophysics is plasma but, naively, one would assume that this introduces both $\mathbf{E}$ and $\mathbf{B}$ fields on an equal footing.

I work in astrophysics and have wondered this from time to time myself. I like the first answer; maybe it also has to do with special relativity telling us that $E$ and $B$ are essentially the same entity? (Just change your reference frame and one becomes another).
–
o0BlueBeast0oJul 8 '14 at 12:34

5 Answers
5

Many astrophysical plasmas are well modeled as perfect conductors. Ideal MHD assumes this limit. As a result, there is no electric field in the fluid's rest frame. In other frames, we generally have $\vec{E} = -\vec{v} \times \vec{B}$, so there is an electric field. However, the perfect conductivity constraint means we don't have to model the electric field - if we evolve just the magnetic field (and the other properties of the fluid like its velocity and density), then we have the complete picture.

The natural followup question is, "Why can we assume infinite conductivity?" Most people's intuition about space is that it is mostly vacuum, and vacuum seems like as good an insulator as one can find. The thing about vacuum is that even though there are few charge carriers per unit volume, what charge carriers there are can proceed uninterrupted and respond to any electric field.

The book Physics of the Interstellar and Intergalactic Medium (Bruce Draine) gives some equations to quantify this. In eq. 35.48 it gives the conductivity of a pure hydrogen fully ionized plasma at temerature $T$ as
$$ \sigma = 4.6\times10^{9}\ \mathrm{s}^{-1} \left(\frac{T}{100\ \mathrm{K}}\right)^{3/2} \left(\frac{30}{\log\Lambda}\right) $$
(CGS units), where kinetic effects and the Debye length are approximately captured by the Coulomb logarithm
$$ \log\Lambda = 22.1 + \log\left(\left(\frac{E}{kT}\right) \left(\frac{T}{10^4\ \mathrm{K}}\right)^{3/2} \left(\frac{n_e}{\mathrm{cm}^{-3}}\right)^{-1}\right) $$
(eq. 2.17). Here $E$ is the particle kinetic energy, and $n_e$ is the number density of electrons.

To give some sense to these numbers, take a look at the conductivities in this table on Wikipedia. Copper has a conductivity of about $6.0\times10^7\ \mathrm{S/m} = 5.4\times10^{17}\ \mathrm{s}^{-1}$, so a $100\ \mathrm{K}$ hydrogen plasma is not nearly as conductive. However, drinking water has a conductivity of no more than $5\times10^{-2}\ \mathrm{S/m} = 4.5\times10^{8}\ \mathrm{s}^{-1}$, and air's conductivity is at most $8\times10^{-15}\ \mathrm{S/m} = 7\times10^{-5}\ \mathrm{s}^{-1}$. Thus astrophysical plasmas are not particularly insulating.

Bruce Draine's book also quotes a timescale for a magnetic field to decay over a lengthscale $L$:
$$ \tau = 5\times10^{8}\ \mathrm{yr}\ \left(\frac{T}{100\ \mathrm{K}}\right)^{3/2} \left(\frac{30}{\log\Lambda}\right) \left(\frac{L}{\mathrm{AU}}\right)^2 $$
(eq. 35.49). Thus if the smallest length scales in your problem are at least $10\ \mathrm{AU}$ (and you are working around $100\ \mathrm{K}$), the magnetic field decay time due to the finite conductivity of the plasma (due in turn to e.g. ion collisions) is well over the current age of the universe. On smaller scales you may have to model such effects, and indeed many astrophysicists do just that.

The lack of the electric field in modeling plasmas stems from the Lorentz force,
$$
\mathbf F=q\mathbf E+q\boldsymbol\beta\times\mathbf B
$$
where $\boldsymbol\beta=\mathbf v/c$. For most astrophysical plasmas, the force is zero, so we have that
$$
\mathbf E=-\boldsymbol\beta\times\mathbf B
$$
So any time we see an electric field, we can simply replace it with the above cross product. This happens in Faraday's law:
$$
\frac{\partial\mathbf B}{\partial t}=-\nabla\times\mathbf E=-\nabla\times\left(-\boldsymbol\beta\times\mathbf B\right)=\nabla\times\boldsymbol\beta\times\mathbf B
$$

The justification for $\mathbf F=0$ is as Chris White says: we assume perfect conductivity.

This has a three-part answer. The first part concerns large-scale, quasi-static electric fields. The nice thing about electric fields is that one can always do a simple galilean transformation to a frame where this quasi-static electric field does not exist. So that is the first part (not very satisfactory, but true and practical).

The second part concerns fluctuating electric fields or small-scale fields. Electric fields do work in plasmas to effectively remove themselves. For instance, imagine that you took a sheet of electrons and a sheet of protons and separated them by some arbitrary distance in a vacuum. Once you remove the mechanical force separating the two charged sheets, the electric fields will do work to remove the charge separation. Granted, if we ignored any deflections, one could imagine these two sheets oscillating back-and-forth ad infinitum. Such oscillations are called plasma oscillations or the simplest form of Langmuir waves. In general, free energy (e.g., non-Maxwellian velocity distributions) in plasmas is removed by exciting instabilities (e.g., leads to the radiation of electromagnetic waves) that then act to further remove free energy.

An example of free energy would be an electron beam in an otherwise isotropic Maxwellian plasma composed of equal number densities of protons and electrons. If the beam moves fast enough relative to the electron thermal speed and has a large enough density relative to the background density, it can excite instabilities that radiate waves (e.g., Langmuir-like oscillations) that act to slow (relative to the background bulk flow) and scatter the beam. Thus, the electric fields of the radiated waves act by doing work on the system in an attempt to return the system to a quasi-equilibrium state.

The third part is somewhat disappointing, in a way. Electric fields introduce an extra order of complexity to an already overly complex system. So it is often desirable to try and work with a system where they do not exist. Computational capacities are limited and so trying to model astrophysical systems with electric fields adds (at least) three extra equations to any simulation. We are only recently reaching the point where we can run 3D particle-in-cell (PIC) simulations of small systems, but even those have limitations (e.g., use of unrealistic mass ratios, M$_{i}$/m$_{e}$) in order to reduce the computations necessary to test any given phenomena.

I think this answer is only partially correct. A star is neutral because the gravitational force is to weak to overcome the electrostatic force between to equally charged particles, not because an atom is neutral.
–
NoldigJul 8 '14 at 12:36

1

voted -1 for it doesn't address the question. It doesn't explain why magnetic effects take over; one could equally as well say that since there are no magnetic monopoles, the atom as a whole is also magnetically neutral.
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Arash ArabiJul 8 '14 at 16:24

1

The atom being electrically neutral ($n_p=n_e$) leads to the large scale neutral state because the plasmas are generally hot enough to consist of only ions & electrons. Note also that local charges can (and do) build up (see any "particle in cell" (PIC) simulation).
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Kyle KanosJul 9 '14 at 14:25

There exists a basic asymmetry: there are no magnetic monopoles of the number and dimensions that the electric monopoles exist ( there are theories with magnetic monopoles and people are looking for them but we are talking of masses much larger than electrons and quarks) .

Hand waving, (because I have not checked the math just extending the symmetry that would have to be imposed in maxwell's equations) , if there existed magnetic monopoles of the size/strength of electric ones ( electrons, quarks) then an alternate state of neutral (magnetically neutral) matter could have appeared, where the spill over electric dipole forces would dominate also over magnetically neutral matter the way magnetic forces do over electrically neutral matter, symmetrically. Then it would be relevant for astrophysics plasma observations, but this is science fiction.